# Gleason's theorem Gleason's theorem In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. André M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.

Conteúdo 1 Declaração do teorema 1.1 Conceptual background 1.2 Deriving the state space and the Born rule 2 History and outline of Gleason's proof 3 Implicações 3.1 Variáveis ​​ocultas 3.2 Quantum logic 4 Generalizações 5 Notas 6 References Statement of the theorem Part of a series of articles about Quantum mechanics {displaystyle ihbar {fratura {parcial }{partial t}}|psi (t)chocalho ={chapéu {H}}|psi (t)chocalho } Schrödinger equation IntroductionGlossaryHistory show Background show Fundamentals show Experiments show Formulations show Equations show Interpretations hide Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning show Scientists vte Conceptual background In quantum mechanics, each physical system is associated with a Hilbert space. For the purposes of this overview, the Hilbert space is assumed to be finite-dimensional. In the approach codified by John von Neumann, a measurement upon a physical system is represented by a self-adjoint operator on that Hilbert space sometimes termed an "observable". The eigenvectors of such an operator form an orthonormal basis for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis. A density operator is a positive-semidefinite operator on the Hilbert space whose trace is equal to 1. In the language of von Weizsäcker, a density operator is a "catalogue of probabilities": for each measurement that can be defined, the probability distribution over the outcomes of that measurement can be computed from the density operator. The procedure for doing so is the Born rule, Que afirma que {estilo de exibição P(x_{eu})=nome do operador {Tr} (Pi _{eu}rho ),} Onde {estilo de exibição rho } is the density operator, e {displaystyle Pi _{eu}} is the projection operator onto the basis vector corresponding to the measurement outcome {estilo de exibição x_{eu}} .

The Born rule associates a probability with each unit vector in the Hilbert space, in such a way that these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis. Além disso, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like a choice of basis for that vector to be embedded in. Gleason's theorem establishes the converse: all assignments of probabilities to unit vectors (ou, equivalentemente, to the operators that project onto them) that satisfy these conditions take the form of applying the Born rule to some density operator. Gleason's theorem holds if the dimension of the Hilbert space is 3 ou melhor; counterexamples exist for dimension 2.

Deriving the state space and the Born rule The probability of any outcome of a measurement upon a quantum system must be a real number between 0 e 1 inclusive, and in order to be consistent, for any individual measurement the probabilities of the different possible outcomes must add up to 1. Gleason's theorem shows that any function that assigns probabilities to measurement outcomes, as identified by projection operators, must be expressible in terms of a density operator and the Born rule. This gives not only the rule for calculating probabilities, but also determines the set of possible quantum states.